Modal Analysis of a AA class Violoncello spruce top plate
This live script presents the complete workflow followed by the authors in the process of determining the dynamic properties of a Violoncello top plate.
After making use of Matlab's own example on the modal analysis of a flexible wing aircraft for the development of this code [1], the authors felt it necessary to share this example in the hopes that it would help others seeking to perform experimental modal analysis through Matlab. Said example was born as a result of one of the authors master's thesis which focused on the manufacture of classical instruments using advanced materials. And was made possible thanks to the help of spanish luthier Carlos Moreno (www.luthieropera.com) who lended the detached top plate used during tests.
As a consequence, the dynamic properties of the currently accepted as a "good" instrument were measured and set as the goal to reach using advanced design and manufacturing techniques. The modal assurance criterion is seen as a way of comparison between not only the different test results but also with test from a simulation.
Another extra feature is the capability of this code of plotting the mode shape using the entire set of 3D mesh points, although only 16 of them had been measured. This is achieved through polynomial fitting on the mode shape components and is application to the rest of the plate's geometry. The authors wish to note that the method used here is only valid for 2D geometries and think that a nice future work would be to extend this to any geometry.
The Violoncello
Parts
Although the Violoncello is comprised of a series of components, all of them involved in the generation of sound (See "the physics of the violin" by Lothar Cremer for further reference), the majority of the sound pressure waves emanate from the top plate-f holes. Therefore, in order to recreate the instrument's characteristic timbre using non-conventional materials dynamic properties of the top plate should be replicated as a minimum.
Obtaining 3D geometry
A 3D scan of both sides of the top plate was performed using an affordable secondhand scanning device. After some treatment of both surfaces, a merger between them was possible creating a 3D geometry.
Creating a Mesh
A mesh was then associated to the 3D geometry using freeware and imposing symmetry between the left and right halves of the geometry. The mesh associated with the longitudinal reinforcement was created separately and restrictions on displacements between them both were imposed.
Test DOFs location
Using other parametric CAD software an array of 16 test points was selected ensuring their symmetric placement.
A template for DOF location on the top plate was made by printing a 2d projection of said mesh.
Test setup
Accelerometers placing
Using the wax provided by the accelerometer manufacturer, accelerometers were attached to the top plate and their DOF numbers were marked on the cables.
Hardware / software
A NI 9185 chassis was used together with NI 9234 data acquisition cards for both the accelerometers and the impact hammer signals. The selected accelerometers had a nominal sensitivity of 100mV/g (although actual calibrated sensitivities were input into the NImax task). and the impact hammer had the lowest possible mass (250gr) and a rubber tip in order to minimize damage to the plate's surface. All test files were stored as ascii formatted data with headers and were converted into .mat format for the sake of simplicity of this live script.
The instrumented plate was hung from a structure using very flexible rubber bands in order to ensure close to free-free boundary conditions during the tests. And the posterior parts of the accelerometers were marked with tape in order to impact on the positive sense of each accelerometer by hitting the plate from its rear side.